Description

Problem 1
(10 points)
Find

Problem 2
(10 points)
a) Show that the volume enclosed when revolving the curve y = f(x) – where f : [a,b] → [0,∞) – about the x-axis in three-dimensional x-y-z space is given by

Hint: Think about the cross-sectional areas and the perfect symmetry when revolving the function around the x-axis. (5 points) b) Compute the volume of the solid obtained by revolving the graph of of on
[0,1] about the x-axis. (5 points)
Problem 3
(10 points)
a) Hook’s law states that the force exerted by an ideal spring when extended from its equi-

librium position at x = 0 to length x is given by
F(x) = −k x,
where k is a positive constant characterizing the stiffness of the spring. Compute the work required to expand the spring from its equilibrium position to length `. (5 points)

b) Show that

is convergent. (5 points)
Hint: There is no elementary way to evaluate this integral. However, to only test convergence, you can bound the integrand by a simpler function and use the following fact without proof: Let f : [a,∞) →R be a bounded and increasing function. Then limx→∞ f(x) exists. (The whole integral corresponds to the function f here.)